Abstract
Recently, Ghosh and Haynes (J Reine Angew Math 712:39–50, 2016) proved a Khintchine-type result for the problem of Diophantine approximation in certain projective spaces. In this note we complement their result by observing that a Jarník-type result also holds for ‘badly approximable’ points in real projective space. In particular, we prove that the natural analogue in projective space of the classical set of badly approximable numbers has full Hausdorff dimension when intersected with certain compact subsets of real projective space. Furthermore, we also establish an analogue of Khintchine’s theorem for convergence relating to ‘intrinsic’ approximation of points in these compact sets.
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Acknowledgments
We would like to thank Prof. Victor Beresnevich and Simon Kristensen for many useful discussions, and also the referee for some useful comments. In addition, the first author believes a paper is never complete without reserved thanks for Prof. Sanju Velani.
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Stephen Harrap: Research supported by EPSRC Grant Number EP/L005204/1 and the Danish research council.
Mumtaz Hussain: Research supported by the Australian research council.
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Harrap, S., Hussain, M. A note on badly approximabe sets in projective space. Math. Z. 285, 239–250 (2017). https://doi.org/10.1007/s00209-016-1705-y
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DOI: https://doi.org/10.1007/s00209-016-1705-y